Fundamental period of `f(x)=sec(sin x)` is

Options

A. `(pi)/2`
B. `2pi`
C. `pi`
D. aperiodic

Explanation

To find the fundamental period of the function \( f(x) = \sec(\sin x) \), we will follow these steps:

### Step 1: Identify the period of the inner function
The inner function here is \( \sin x \). The period of \( \sin x \) is \( 2\pi \).

**Hint:** The period of a sine function is always \( 2\pi \).

### Step 2: Check for a smaller period
We will check if the function \( f(x) \) has a smaller period, specifically \( \pi \). For a function to be periodic with period \( p \), it must satisfy the condition \( f(p + x) = f(x) \).

**Hint:** To check for periodicity, we need to evaluate \( f(\pi + x) \) and see if it equals \( f(x) \).

### Step 3: Compute \( f(\pi + x) \)
We compute:
\[ f(\pi + x) = \sec(\sin(\pi + x)) \]
Using the property of sine, we know that:
\[ \sin(\pi + x) = -\sin(x) \]
Thus, we can rewrite:
\[ f(\pi + x) = \sec(-\sin(x)) \]

**Hint:** Remember that \( \sec(-x) = \sec(x) \).

### Step 4: Simplify \( f(\pi + x) \)
Using the property of secant:
\[ \sec(-\sin(x)) = \sec(\sin(x)) \]
Therefore, we have:
\[ f(\pi + x) = \sec(\sin(x)) = f(x) \]

**Hint:** If \( f(p + x) = f(x) \), then \( p \) is a period of the function.

### Step 5: Conclusion
Since we have shown that \( f(\pi + x) = f(x) \), we conclude that the fundamental period of \( f(x) = \sec(\sin x) \) is \( \pi \).

**Final Answer:** The fundamental period of \( f(x) = \sec(\sin x) \) is \( \pi \).